Ch 42: Nuclear Physics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 42: Nuclear Physics

Problem 56

Chapter 42, Problem 56

A sample contains radioactive atoms of two types, A and B. Initially there are five times as many A atoms as there are B atoms. Two hours later, the numbers of the two atoms are equal. The half-life of A is 0.50 hour. What is the half-life of B?

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Define the decay law for radioactive atoms: The number of atoms remaining at time \( t \) is given by \( N(t) = N_0 e^{-\lambda t} \), where \( N_0 \) is the initial number of atoms, \( \lambda \) is the decay constant, and \( t \) is the time elapsed.

Relate the decay constant \( \lambda \) to the half-life \( T_{1/2} \) using the formula \( \lambda = \frac{\ln(2)}{T_{1/2}} \). For atom A, \( \lambda_A = \frac{\ln(2)}{0.50} \). For atom B, \( \lambda_B = \frac{\ln(2)}{T_{1/2,B}} \), where \( T_{1/2,B} \) is the unknown half-life of B.

Set up the initial condition: Let the initial number of B atoms be \( N_{B0} \). Then the initial number of A atoms is \( N_{A0} = 5N_{B0} \), as given in the problem.

Write the decay equations for the number of atoms after 2 hours: \( N_A(2) = N_{A0} e^{-\lambda_A \cdot 2} \) and \( N_B(2) = N_{B0} e^{-\lambda_B \cdot 2} \). Since the numbers of A and B atoms are equal after 2 hours, set \( N_A(2) = N_B(2) \).

Substitute the expressions for \( N_A(2) \) and \( N_B(2) \) into the equality: \( 5N_{B0} e^{-\lambda_A \cdot 2} = N_{B0} e^{-\lambda_B \cdot 2} \). Cancel \( N_{B0} \) and solve for \( \lambda_B \) in terms of \( \lambda_A \). Finally, use \( \lambda_B = \frac{\ln(2)}{T_{1/2,B}} \) to find \( T_{1/2,B} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This decay occurs at a characteristic rate for each radioactive isotope, defined by its half-life, which is the time required for half of the radioactive atoms in a sample to decay. Understanding this concept is crucial for analyzing the changes in the number of radioactive atoms over time.

Half-Life

The half-life of a radioactive substance is the time it takes for half of the original amount of the substance to decay. For example, if a sample has a half-life of 0.50 hours, after this time, only half of the original atoms will remain. This concept is essential for calculating the remaining quantities of radioactive isotopes after a given period, as seen in the problem involving atoms A and B.

Exponential Decay

Exponential decay describes the process where a quantity decreases at a rate proportional to its current value. In the context of radioactive decay, the number of atoms decreases exponentially over time, which can be mathematically represented by the equation N(t) = N0 * (1/2)^(t/T), where N0 is the initial quantity, t is time, and T is the half-life. This concept is vital for determining the relationship between the quantities of atoms A and B after a specified time.

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